Revealing short-range non-Newtonian gravity through Casimir–Polder shielding

Our envisaged experiment features an atom interacting with a multi-layered medium consisting of vacuum-gold-vacuum-silicon, as shown in figure 3. In order to have a tractable calculation of the CP force for this situation we will make two simplifying approximations; (1) The silicon slab will be modelled as having infinite depth, and (2) we assume that all layers are of infinite extent in the lateral directions.

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Let us comment on these simplifications. The first approximation will over-estimate the CP force because as long as the depth of the silicon slab is significantly larger than the atomic transition wavelength (we will choose parameter values satisfying this condition), the CP force from any finite-depth slab will be smaller than that for an infinitely deep one. Note that at the atomic transition wavelength considered here, silicon has quite a large skin depth of δ ∼ 17 μm⁴ so the over-estimation can be quite large for thin slabs. The second approximation, meanwhile, is accurate if the lateral dimensions of the apparatus are much larger than all other length scales. Like the first approximation, assuming infinite lateral size will over-estimate the magnitude of the CP force. Therefore, our approximation scheme will give a reliable upper bound for the CP force. Since the CP force represents an unwanted effect when measuring gravity, an upper bound is still of value. By contrast, we shall not make these approximations in our calculation of the Yukawa force to be described in section 2.

The CP potential arises from the interaction energy between a fluctuating dipole and a nearby macroscopic body. The coupling strengths involved are the polarisability of the atom α(ω) and the reflectivity of the macroscopic body, which for a planar surface is encoded in the transverse electric (TE) and transverse magnetic (TM) reflection coefficients r_TE and r_TM. The interaction is mediated by photons of wave vector ${\bf{k}}$, which in a planar system is decomposed into parallel (${k}_{\parallel }$) and perpendicular components

$$\begin{eqnarray}&&{k}_{\perp }=\sqrt{{\left(\omega /c\right)}^{2}-{k}_{\parallel }^{2}}.\end{eqnarray} \tag{ 2 }$$

Summing over all such photon contributions entails integrating over any two of the three variables $\omega ,{k}_{\parallel },{k}_{\perp }$, since the third is fixed by equation (2). Here we choose to eliminate ${k}_{\perp }$. The final step is to rotate the ω-integration to imaginary frequencies ω → iξ (ξ > 0) in order to avoid a rapidly oscillating integrand. Putting this all together one finds the CP potential U_CP a distance z from a planar surface [17–19]

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{U}_{{\rm{CP}}}(z) & = & \displaystyle \frac{\hslash {\mu }_{0}}{8{\pi }^{2}}{\displaystyle \int }_{0}^{\infty }{\rm{d}}\xi \,{\xi }^{2}\alpha ({\rm{i}}\xi ){\displaystyle \int }_{0}^{\infty }{{\rm{d}}k}_{\parallel }\displaystyle \frac{{k}_{\parallel }}{\sqrt{{k}_{\parallel }^{2}+{\xi }^{2}/{c}^{2}}}\times \left[{r}^{{\rm{TE}}}-\left(1+2\displaystyle \frac{{k}_{\parallel }^{2}{c}^{2}}{{\xi }^{2}}\right){r}^{{\rm{TM}}}\right]{{\rm{e}}}^{-2z\sqrt{{k}_{\parallel }^{2}+{\xi }^{2}/{c}^{2}}},\end{array}\end{array}\end{eqnarray} \tag{ 3 }$$

where the polarizability α(ω) is defined for a transition i → j of an isotropically polarisable atom as:

$$\begin{eqnarray}&&\alpha (\omega )=\displaystyle \frac{2}{3{\hslash }}\mathop{\mathrm{lim}}\limits_{\epsilon \to 0}\displaystyle \frac{{\omega }_{{ij}}| {\mu }_{{ij}}{| }^{2}}{{\omega }_{{ij}}^{2}-{\omega }^{2}-{\rm{i}}\omega \epsilon }.\end{eqnarray} \tag{ 4 }$$

Here, μ_ij is the atomic transition dipole moment and ω_ij is the transition frequency.

We can build the explicit reflection coefficients for either polarisation $\sigma =\mathrm{TE}$, TM by beginning from the well-known expressions for the overall reflection coefficient ${r}_{{ijk}}^{\sigma }$ of a three-layer system whose two interfaces are separated by a distance d_j and have single-interface Fresnel reflection coefficients ${r}_{{ij}}^{\sigma }$ and ${r}_{{jk}}^{\sigma }$, and single-interface transmission coefficients ${t}_{{ij}}^{\sigma }$ and ${t}_{{jk}}^{\sigma }$ as listed in the appendix. The composite reflection coefficients are given by [20]

$$\begin{eqnarray}&&{r}_{{ijk}}^{\sigma }={r}_{{ij}}^{\sigma }+\displaystyle \frac{{t}_{{ij}}^{\sigma }{t}_{{ji}}^{\sigma }{r}_{{jk}}^{\sigma }{{\rm{e}}}^{2{\rm{i}}{\beta }_{j}{d}_{j}}}{1-{r}_{{ji}}^{\sigma }{r}_{{jk}}^{\sigma }{{\rm{e}}}^{2{\rm{i}}{\beta }_{j}{d}_{j}}},\end{eqnarray} \tag{ 5 }$$

where ${\beta }_{i}=\sqrt{{\varepsilon }_{i}{\omega }^{2}/{c}^{2}-{k}_{\parallel }^{2}}$ is the z-component of the wave vector in layer i. In our particular system, we have;

$$\begin{eqnarray}&&{\beta }_{1}={\beta }_{3}={\beta }_{\mathrm{vac}}=\sqrt{{\omega }^{2}/{c}^{2}-{k}_{\parallel }^{2}},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\beta }_{2}={\beta }_{\mathrm{Au}}=\sqrt{{\varepsilon }_{\mathrm{Au}}{\omega }^{2}/{c}^{2}-{k}_{\parallel }^{2}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\beta }_{4}={\beta }_{\mathrm{Si}}=\sqrt{{\varepsilon }_{\mathrm{Si}}{\omega }^{2}/{c}^{2}-{k}_{\parallel }^{2}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{d}_{1}=z,\quad {d}_{2}={d}_{\mathrm{Au}},\quad {d}_{3}={d}_{\mathrm{vac}},\quad {d}_{4}\to \infty ,\end{eqnarray} \tag{ 9 }$$

where ε_Au and ε_Si are the relative permittivities of gold and silicon, respectively. This means that r₂₃₄ as shown in figure 3 is given by

$$\begin{eqnarray}&&{r}_{234}^{\sigma }={r}_{23}^{\sigma }+\displaystyle \frac{{t}_{23}^{\sigma }{t}_{32}^{\sigma }{r}_{34}^{\sigma }{{\rm{e}}}^{2{\rm{i}}{\beta }_{\mathrm{vac}}{d}_{\mathrm{vac}}}}{1-{r}_{32}^{\sigma }{r}_{34}^{\sigma }{{\rm{e}}}^{2{\rm{i}}{\beta }_{\mathrm{vac}}{d}_{\mathrm{vac}}}}.\end{eqnarray} \tag{ 10 }$$

Now we can build the reflection coefficient of the whole system by composing r₂₃₄ with r₁₂:

$$\begin{eqnarray}&&{r}_{1234}^{\sigma }={r}_{12}^{\sigma }+\displaystyle \frac{{t}_{12}^{\sigma }{t}_{21}^{\sigma }{r}_{234}^{\sigma }{{\rm{e}}}^{2{\rm{i}}{\beta }_{\mathrm{Au}}{d}_{\mathrm{Au}}}}{1-{r}_{21}^{\sigma }{r}_{234}^{\sigma }{{\rm{e}}}^{2{\rm{i}}{\beta }_{\mathrm{Au}}{d}_{\mathrm{Au}}}}.\end{eqnarray} \tag{ 11 }$$

The explicit form of these composite reflection coefficients can be found in the appendix. The CP potential of the atom in our particular setup is then finally given by

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{U}_{{\rm{CP}}} & = & \displaystyle \frac{\hslash {\mu }_{0}}{8{\pi }^{2}}{\displaystyle \int }_{0}^{\infty }{\rm{d}}\xi \,{\xi }^{2}\alpha ({\rm{i}}\xi ){\displaystyle \int }_{0}^{\infty }{{\rm{d}}k}_{\parallel }\displaystyle \frac{{k}_{\parallel }}{\sqrt{{k}_{\parallel }^{2}+{\xi }^{2}/{c}^{2}}}\times \left[{r}_{1234}^{{\rm{TE}}}-\left(1+2\displaystyle \frac{{k}_{\parallel }^{2}{c}^{2}}{{\xi }^{2}}\right){r}_{1234}^{{\rm{TM}}}\right]{{\rm{e}}}^{-2z\sqrt{{k}_{\parallel }^{2}+{\xi }^{2}/{c}^{2}}}.\end{array}\end{array}\end{eqnarray} \tag{ 12 }$$

This expression allows us to include the effects of finite thickness and conductivity of the shield on the CP force. Note that if the gold layer was a perfect reflector we could set ${r}_{1234}^{\mathrm{TE}}\to -1$ and ${r}_{1234}^{\mathrm{TM}}\to 1$.

The main contribution to the CP potential is at the dominant transition frequency of the atom, which for ground state Rb atoms is approximately 3.8 × 10¹⁴ Hz (780 nm). We propose choosing a gold shield of thickness d_Au = 50 nm which is considerably greater than the skin depth at this frequency which is of the order of a few nanometres. As shown in the upper plot of figure 4, it is indeed the case that by the time the gold is 50 nm thick the CP force is rather insensitive to its finite thickness. However, as well as having a finite thickness the gold also has a finite conductivity. This is well-described by an intraband Drude model whose dielectric function is:

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{Au}}(\omega )=1-\displaystyle \frac{{\omega }_{p}^{2}}{\omega (\omega +{\rm{i}}\gamma )},\end{eqnarray} \tag{ 13 }$$

where ${\omega }_{p}=1.38\times {10}^{16}$ rad s⁻¹ is the plasma frequency and γ ≈ 4.08 × 10¹³ rad s⁻¹ is the damping parameter [21]. In principle one should also take into account interband transitions (see, e.g. [22]), but these only have an effect on the permittivity well above the frequency scale set by the considered transition of the atom, meaning they have a negligible effect on the results presented here. This finite conductivity has an appreciable effect on the CP force near the sheet of gold (as compared to a perfect reflector), as shown in the lower panel of figure 4. This suggests that in order to get reliable results for the relative values of the various forces involved a realistic CP calculation, as performed here, is necessary.

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We now proceed to a calculation of the Yukawa force. This is much simpler than that of the CP force and we therefore include both the finite depth of the silicon slab and the finite lateral sizes of each layer. Referring back to figure 2, the values we shall use for the various parameters are a = 100 μm, b = 100 μm and W = 10 μm, and as above we assume the thickness of the gold sheet to be d_Au = 50 nm.

In order to calculate the Yukawa force in the system we require an expression for it at a distance $Z={d}_{\mathrm{vac}}+{d}_{\mathrm{Au}}+z+W/2$ from the centre of mass of a rectangular slab, as shown in figure 2. The force is evaluated on the xy symmetry axis of the slab so symmetry dictates that the force ${{\bf{F}}}_{{\rm{Y}}}$ is in the ${\bf{z}}$ direction; ${{\bf{F}}}_{{\rm{Y}}}=\left[\tfrac{{\rm{d}}}{{\rm{d}}{Z}}{U}_{{\rm{Y}}}(r)\right]\hat{{\bf{Z}}}=\left[\tfrac{{\rm{d}}}{{\rm{d}}{z}}{U}_{{\rm{Y}}}(r)\right]\hat{{\bf{z}}}$. We can then integrate this over the slab volume to obtain the Yukawa force from the whole slab. We have

$$\begin{eqnarray}&&{{\bf{F}}}_{{\rm{Y}}}=(Z\hat{{\bf{z}}}-{\bf{r}})\displaystyle \frac{{Gm}\rho \alpha }{\lambda }{\int }_{V}{{\rm{d}}}^{3}{\bf{r}}\displaystyle \frac{| Z\hat{{\bf{z}}}-{\bf{r}}| +\lambda }{| Z\hat{{\bf{z}}}-{\bf{r}}{| }^{3}}{{\rm{e}}}^{-| {\bf{r}}-Z\hat{{\bf{z}}}| /\lambda },\end{eqnarray} \tag{ 14 }$$

where V is the volume of the slab and ρ is its density. Taking initially the case of a slab of infinite extent in the xy plane and of thickness W, we find, in agreement with [23], an exact result for the Yukawa force in such a situation

$$\begin{eqnarray}&&{{\bf{F}}}_{{\rm{Y}}}^{\inf }=4\pi \alpha G\lambda m\rho {{\rm{e}}}^{-Z/\lambda }\sinh \left(\displaystyle \frac{W}{2\lambda }\right)\hat{{\bf{z}}}.\end{eqnarray} \tag{ 15 }$$

For a finite slab the integrals must be carried out numerically, which is complicated by the fact that the parameters α and λ are unknown. One can eliminate the overall scale α by expressing the finite slab force in units of the infinite slab force, but the range parameter λ remains. The results for various values of λ of a finite-slab numerical integration are shown in figure 5.

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Now that we have an account of the Yukawa and CP forces involved in the proposed setup, we can compare them to determine which is dominant and whether any are amenable to measurement. The forces due to each component of the apparatus (and also the Earth) are listed in table 1 and are plotted in figure 6 as a function of ${d}_{\mathrm{vac}}$ (the two rows in figure 6 are for two different fixed values of z). The parameters we use are given in table 2 and correspond to the four points chosen in figure 1. In the plot we have graphed both the absolute values of the forces and the quantity

$$\begin{eqnarray}&&{\rm{\Delta }}F=F-F({d}_{\mathrm{Vac}}\to \infty )\end{eqnarray} \tag{ 16 }$$

so that, for example, the ΔF for the Yukawa force of the silicon slab is

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{{\rm{Y}}(\mathrm{Si})}={F}_{{\rm{Y}}(\mathrm{Si})}-{F}_{{\rm{Y}}(\mathrm{Si})}({d}_{\mathrm{Vac}}\to \infty )={F}_{{\rm{Y}}(\mathrm{Si})}\end{eqnarray} \tag{ 17 }$$

and for the gold shield

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{{\rm{Y}}(\mathrm{Au})}={F}_{{\rm{Y}}(\mathrm{Au})}-{F}_{{\rm{Y}}(\mathrm{Au})}({d}_{\mathrm{Vac}}\to \infty )=0\end{eqnarray} \tag{ 18 }$$

and so on. By subtracting out the value at ${d}_{\mathrm{vac}}=\infty$, ΔF gives us the difference between the cases where the silicon slab is present and when it is removed.

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Table 1.  Summary of forces involved. Here $Z={d}_{\mathrm{vac}}+{d}_{\mathrm{Au}}+z+W/2$.

Symbol	Description	Force found from
Yi(Si)	Yukawa force of the slab with parameters i = 1, 2, 3, 4 defined in figure 1	$4\pi {\alpha }_{i}G\lambda {m}_{\mathrm{Rb}}{\rho }_{\mathrm{Si}}{{\rm{e}}}^{-Z/{\lambda }_{i}}\sinh \left(\tfrac{W}{2{\lambda }_{i}}\right)$
Yi(Au)	As above, but for the shield	$4\pi {\alpha }_{i}G{\lambda }_{i}{m}_{\mathrm{Rb}}{\rho }_{\mathrm{Au}}{{\rm{e}}}^{-{z}/{\lambda }_{i}}\sinh \left(\tfrac{{d}_{\mathrm{Au}}}{2{\lambda }_{i}}\right)$
CP	Casimir–Polder force in the presence of the gold shield, including non-additive contributions from both the slab and the shield	Numerical evaluation of equation (3)
CP(Si)	Casimir–Polder force if the shield is removed	Numerical evaluation of equation (3) with εAu(ω) = 1
N(Si)	Newtonian gravitational force of the slab	${GabW}{\rho }_{\mathrm{Si}}{m}_{\mathrm{Rb}}/{Z}^{2}$
N(Au)	Newtonian gravitational force of the shield	$2\pi G{\rho }_{\mathrm{Au}}{m}_{\mathrm{Rb}}$
E	Gravitational force of the Earth	${m}_{\mathrm{Rb}}g$

Table 2.  Summary of parameters used in this work.

Yukawa		Slab		Shield		Atom
$\{\alpha ,\lambda \}{}_{1}$	$\{{10}^{9},2\,\mu {\rm{m}}\}$	ρSi	2 330 kg m−3	ρAu	19 300 kg m−3	μij	5.05 × 10−29 C m
{α, λ }2	$\left\{{10}^{6},2\,\mu {\rm{m}}\right\}$	εSi	5	ωp	1.38 × 1016 rad s−1	ωij	2.4 × 1015 rad s−1
${\left\{\alpha ,\lambda \right\}}_{3}$	$\left\{{10}^{9},0.5\,\mu {\rm{m}}\right\}$	W	10 μm	γ	4 × 1013 rad s−1	mRb	1.4 × 10−25 kg
{α, λ}4	{106,0.5 μ m}	a, b	100 μm	dAu	50 nm

From the left-hand column of figure 6 we see that the Newtonian gravitational force of the Earth (m_Rb g) dominates all other forces in all cases, as one would expect. By contrast, we see from the right column of figure 6, where ΔF is plotted, that the dominant change in the CP contribution with and without the silicon slab is the Yukawa force corresponding to points 1 and 2 on figure 1. The Earth's gravity and the various shield forces obviously do not change before and after the removal of the slab, and so do not appear on the right-hand graphs. The figure also emphasises the change in the CP force due to the presence of the shield, with the improvement to the shielded case highlighted as the shaded area. It is clear from the graphs in the right-hand column that the consideration of realistic shielding of CP forces as given in this paper is necessary to properly evaluate the forces in the proposed system.

The key message conveyed by figure 6 is that if the Yukawa modification to Newtonian gravity exists with any of the parameters chosen in figure 1, then the detected change in the total force ΔF with and without the silicon slab would be much larger than would be expected based on the CP force and Newtonian gravity alone, indicating the presence of a new force. This does not so far say anything about whether the size of that discrepancy is actually detectable; this aspect will be considered in section 4.

We also briefly consider thermal effects. The regime in which thermal effects can become important to the CP forces is when the black-body peak is at a wavelength comparable to the atom-surface distance (see, for example, [24]). At 300 K, this peak is at around 17 μm, so for parameters used in the uppermost graphs in figure 6 (which turn out to be the ones we are mainly interested in) we can safely ignore thermal effects even at room temperature. In order to go to longer distances and still be able to ignore thermal effects one would have to cool the apparatus, but cryogenic temperatures are not necessary. For example, at 150 K the black-body peak is at approximately 34 μm, which easily encompasses the distances we are interested in.

Although our calculations of the forces on an atom detailed above are independent of the measurement method employed, in order to evaluate the viability of the entire scheme presented here let us now examine one particular method, namely, gravity-induced Bloch oscillations [13, 25]. This is a type of atom interferometry and takes advantage of the quantum wave-like properties of atoms. However, unlike standard atom interferometers where interference takes place between different paths in coordinate space, and which have, for example, been used to make high precision measurements of the gravitational constant G [26, 27], Bloch oscillations are the result of interference in momentum space and the motion of the atoms in coordinate space can be made very small. This allows the atoms to be localised at an almost fixed distance from a surface and is well suited to measuring short-range forces [28]. Indeed, referring to the right hand column of figure 6, we see that the Yukawa force diminishes significantly with distance (although it should be noted that ΔF_Y remains dominant all the way out to 30 μm). Thus, we want the atoms to be located as close as possible to the shield.

Bloch oscillations occur when an external force is applied to a quantum particle that also experiences a periodic potential, which in the present case would be provided by an optical lattice formed by retro-reflection of a laser beam from the gold shield. Bloch oscillations are sensitive directly to the force, as opposed to collective dipole oscillations in a harmonic trap [29, 30] (sensitive to force gradients), and so-called super-Bloch oscillations in driven optical lattices (sensitive directly to potential) [31], and can even be measured non-destructively [32–34]. The Bloch oscillation frequency ν_B is directly proportional to the external force F

$$\begin{eqnarray}&&{\nu }_{{\rm{B}}}=\displaystyle \frac{{Fa}}{2\pi {\hslash }},\end{eqnarray} \tag{ 19 }$$

where a is the lattice spacing, which we will take to be 500 nm, but is independent of the depth of the lattice potential. In our proposed experiment the strongest force by some orders of magnitude is ${m}_{\mathrm{Rb}}g\approx 1.4\times {10}^{-24}\,{\rm{N}}$, which corresponds to a Bloch oscillation frequency of around 1 kHz. This fast oscillation can be used to our advantage [28]: orienting the apparatus vertically the 'little g' driven oscillations provide a reference oscillator whose frequency can be measured to an accuracy of one part in 10⁷ [13, 25]. Moving the silicon slab to different positions, Z_i and Z_f say, one can attempt to measure the shift Δν_B in the Bloch oscillation frequency

$$\begin{eqnarray}&&{\rm{\Delta }}{\nu }_{{\rm{B}}}=\displaystyle \frac{[F({Z}_{f})-F({Z}_{i})]}{2\pi {\hslash }}a.\end{eqnarray} \tag{ 20 }$$

The difference in the force for the two positions of the silicon slab is therefore measurable providing Δν_B is larger than around ${10}^{-7}\,\mathrm{kHz}=0.1\,\mathrm{mHz}$.

As mentioned above, we would like to trap the atoms as close to the shield as possible, but in practical terms there is of course a limit to how close to the surface one can go—this is set by how tightly the atoms can be trapped and how far they move during one Bloch oscillation. For discussion of the former we note from [13, 35] that a typical atomic cloud used in Bloch oscillation experiments has an rms width of approximately 12 μm, meaning that trapping distances much closer than this are unrealistic. For example, as shown in the lower part of figure 7, an atomic cloud trapped at 10 μm with an rms width of 12 μm has appreciable overlap with the surface (approximately 2.5% of the atoms would be in the region z < 0). At such a large distance the techniques proposed here would only result in a modest improvement in the region of Yukawa parameter space that can be excluded, as can be seen in figure 7.

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We therefore consider what would happen if, in a future experiment, an atomic cloud could be centred at a distance of 3 μm from the surface with an rms width of 2 μm. As shown in the lower part of figure 7, the overlap referred to above would be much smaller (approximately 0.02%). To this end we note that in deep lattices, or close to the Mott insulator phase [36], atoms occupy close to a single lattice site. In fact, operating in such regimes may not be necessary because the effect of adding an external force on top of a periodic potential is exactly to localise the atoms in space such that the (localised) Wannier–Stark states become the natural eigenfunctions rather than the (delocalised) Bloch functions [37]. During a Bloch oscillation wave packets explore a spatial region of size w, given by [38]

$$\begin{eqnarray}&&w=\displaystyle \frac{W}{2F},\end{eqnarray} \tag{ 21 }$$

where W is the width of the first energy band. In the tight-binding approximation and with the depth of the lattice being five times the photon recoil energy ${E}_{{\rm{R}}}={{\hslash }}^{2}{k}_{{\rm{L}}}^{2}/(2m)$, where k_L is the laser wavenumber, one finds that $W\approx 0.26{E}_{{\rm{R}}}$ [38]. For our parameters we find ${E}_{{\rm{R}}}=6.13\times {10}^{-30}\,{\rm{J}}$, using this in equation (21), one finds w ≈ 0.6 μm, which is a good deal smaller than the proposed trapping distance of 3 μm. This means that the oscillations remain localised around 3 μm, which is the region in which we have shown the Yukawa force change can dominate. Furthermore, one need not rely solely on little g to provide the fast Bloch oscillation frequency and localisation: an external magnetic field interacting with the atoms' magnetic moment (the basis of magnetic traps) can be used to boost the external force which would also improve the accuracy of the frequency measurement by increasing the number of oscillations during the measurement time. A careful consideration of any systematic effects associated with this field would have to be made, although we note that magnetic fields near gold-coated silicon chips have been well studied in the context of atom chips (see, e.g. [39]).

Finally, we consider which parts of the parameter space of the Yukawa interaction could be excluded in an experiment such as the one we propose here. Firstly we fix z to 3 μm based on the discussion above, giving us a value for ΔF_CP. Equating this to ${\rm{\Delta }}{F}_{{\rm{Y}}(\mathrm{Si})}$ as given by equation (17) with unspecified α and λ, one then has an equation that constrains α and λ to a particular set of combinations represented by a curve in the α–λ plane. The region bounded by this curve contains all the values of α and λ for which the Yukawa force change is larger than the CP force change in this experiment. This region is shown in figure 1, alongside that for the unshielded case (found by solution of ${\rm{\Delta }}{F}_{{\rm{Y}}(\mathrm{Si})}={\rm{\Delta }}{F}_{\mathrm{CP}({\rm{S}})}$) and the real experimental sensitivity (found by solution of ${\rm{\Delta }}{F}_{{\rm{Y}}(\mathrm{Si})}={10}^{-31}\,{\rm{N}}$, which is the force corresponding to a frequency sensitivity of 0.1mHz as discussed above). It is seen that the CP shield is vital if any part of the parameter space is to be excluded by this type of experiment, and that there is a considerably sized new region in which the Yukawa force change overwhelms all others, without consideration of the absolute magnitude (i.e. detectability) of such a force. Taking into account the real experimental sensitivity of a Bloch oscillation experiment, one still finds that a significant new region of parameter space could be excluded.

Here we have demonstrated that a realistic account of the CP force is an important ingredient in the design of experiments using electromagnetic shielding to measure short-range corrections to Newtonian gravity, and that these considerations are in fact the deciding factor in whether the non-Newtonian force can be dominant over all others in a particular experimental setup. Parameterising this force by a Yukawa potential, we have made an initial investigation into whether this force is large enough to be measurable, finding that, given modest improvements in localisation of atoms in an optical lattice, a new region of the Yukawa parameter space can be excluded.

We gratefully acknowledge E A Hinds, J D D Martin and I Yavin for advice and suggestions. DO thanks the Natural Sciences and Engineering Research Council of Canada (NSERC) for funding, and RB thanks the UK Engineering and Physical Sciences Research Council (EPSRC), the Alexander von Humboldt Foundation and Freiburg Institute for Advanced Studies (FRIAS).

The single-interface reflection and transmission coefficients are;

$$\begin{eqnarray}&&{r}_{{ij}}^{\mathrm{TE}}=\displaystyle \frac{{\beta }_{i}-{\beta }_{j}}{{\beta }_{i}+{\beta }_{j}}\qquad {r}_{{ij}}^{\mathrm{TM}}=\displaystyle \frac{{\varepsilon }_{j}{\beta }_{i}-{\varepsilon }_{i}{\beta }_{j}}{{\varepsilon }_{j}{\beta }_{i}+{\varepsilon }_{i}{\beta }_{j}},\end{eqnarray} \tag{ A.1 }$$

$$\begin{eqnarray}&&{t}_{{ij}}^{\mathrm{TE}}=1+{r}_{{ij}}^{\mathrm{TE}}\qquad {t}_{{ij}}^{\mathrm{TM}}=\displaystyle \frac{{\varepsilon }_{i}}{{\varepsilon }_{j}}\left(1+{r}_{{ij}}^{\mathrm{TM}}\right).\end{eqnarray} \tag{ A.2 }$$

Using a shorthand ${S}_{i}={{\rm{e}}}^{2{\rm{i}}{\beta }_{i}{d}_{i}}$, the four layer reflection coefficient used in the text is, for either polarisation;

$$\begin{eqnarray}&&{r}_{1234}^{\sigma }={r}_{12}^{\sigma }-\displaystyle \frac{{S}_{2}({r}_{12}^{\sigma }+1)({r}_{21}^{\sigma }+1)\left[{S}_{3}({r}_{32}^{\sigma }+1){r}_{34}^{\sigma }+{r}_{23}^{\sigma }\left(1+{S}_{3}{r}_{34}^{\sigma }\right)\right]}{{S}_{3}{r}_{32}^{\sigma }{r}_{34}^{\sigma }+{S}_{2}{r}_{21}^{\sigma }\left[{S}_{3}({r}_{32}^{\sigma }+1){r}_{34}^{\sigma }+{r}_{23}^{\sigma }\left(1+{S}_{3}{r}_{34}^{\sigma }\right)\right]-1}\end{eqnarray} \tag{ A.3 }$$

which is a more explicit version of equation (11).